CUET Mathematics - Calculus - Integrals | Free Questions - Page 3

Q1:

2026: 22 May Shift 1

Calculus

>

Integrals

Medium

Applied

$I=\int_{2}^{8}\frac{\sqrt[3]{x+1}}{\sqrt[3]{x+1}+\sqrt[3]{11-x}}dx$ is equal to :

$\frac{1}{2}$

6

10

3

Correct Answer

Option 4

Correct Answer

Explanation

Q2:

2026: 22 May Shift 1

Calculus

>

Integrals

Medium

Applied

$I=\int(\log x)^2dx$ is equal to :(Consider $\log_e x=\log x$ )

$x(\log x)^2+2x\log x+2x+C$ , where C is an arbitrary constant

$x\log x+2x\log x+2x+C$ , where C is an arbitrary constant

$x(\log x)^2-2x\log x+2x+C$ , where C is an arbitrary constant

$x(\log x)^2-2x\log x-2x+C$ , where C is an arbitrary constant

Correct Answer

Option 3

Correct Answer

Explanation

Q3:

2026: 21 May Shift 2

Calculus

>

Integrals

Hard

Common

$\displaystyle\int \dfrac{x + \dfrac{a}{x}}{x^2 + \dfrac{a^2}{x^2}}\,dx, a > 0$ is equal to (where C is an arbitrary constant)

$\dfrac{1}{4}\log_e|x^4 + a^2| + \dfrac{1}{2}\tan^{-1}\dfrac{x^2}{a} + C$

$\dfrac{1}{2}\log_e|x^2 + a| + \dfrac{1}{4}\tan^{-1}\dfrac{x^4}{a^2} + C$

$\dfrac{1}{4}\log_e|x^4 + a^2| + \dfrac{a}{2}\tan^{-1}\dfrac{x^2}{a} + C$

$\dfrac{1}{2}\log_e|x^2 + a| + \dfrac{a}{4}\tan^{-1}\dfrac{x^4}{a^2} + C$

Correct Answer

Option 1

Correct Answer

Explanation

Q4:

2026: 21 May Shift 2

Calculus

>

Integrals

Easy

Common

Match the LIST-I with LIST-II

LIST-I (Definite integral)		LIST-II (Value of definite integral)
A. $\displaystyle\int_4^9 \dfrac{1}{\sqrt{x}}\,dx$		I. $\dfrac{2}{3}$
B. $\displaystyle\int_{-1}^1 x^3\,dx$		II. 1
C. $\displaystyle\int_{-1}^1 x^2\,dx$		III. 0
D. $\displaystyle\int_1^2 [x]\,dx$ , where $[\cdot]$ denotes greatest integer function		IV. 2

Choose the correct answer from the options given below:

A-I, B-II, C-III, D-IV

A-III, B-IV, C-I, D-II

A-IV, B-III, C-I, D-II

A-II, B-III, C-I, D-IV

Correct Answer

Option 3

Correct Answer

Explanation

Q5:

2026: 21 May Shift 2

Calculus

>

Integrals

Medium

Core

$\displaystyle\int e^x(\sec x + \log_e(\sec x + \tan x))dx$ is equal to: (where C is an arbitrary constant)

$e^x\log_e(\tan x) + C$

$e^x\log_e(\sec x) + C$

$e^x\sec x + C$

$e^x\log_e(\sec x + \tan x) + C$

Correct Answer

Option 4

Correct Answer

Explanation

Q6:

2026: 21 May Shift 2

Calculus

>

Integrals

Hard

Core

Let $[t]$ denotes the greatest integer $\le t$ , then the value of $\left[\displaystyle\int_{\pi}^{3\pi/2} x|\sin x|\,dx\right]$ is:

3

4

5

6

Correct Answer

Option 2

Correct Answer

Explanation

Q7:

2026: 21 May Shift 2

Calculus

>

Integrals

Easy

Core

Match the LIST-I with LIST-II

	LIST-I Definite Integrals		LIST-II Values
A.	$\displaystyle\int_{-1}^{1} \vert x \vert \, dx$	I.	$2$
B.	$\displaystyle\int_{-1}^{1} \sin^3 x \cos^4 x \, dx$	II.	$1$
C.	$\displaystyle\int_{-1}^{1} 3\vert x \vert^2 \, dx$	III.	$0$
D.	$\displaystyle\int_{0}^{4} \vert x-1 \vert \, dx$	IV.	$5$

Choose the correct answer from the options given below:

A-II, B-III, C-I, D-IV

A-IV, B-I, C-II, D-III

A-III, B-II, C-IV, D-I

A-II, B-I, C-IV, D-III

Correct Answer

Option 1

Correct Answer

Explanation

Q8:

2026: 21 May Shift 2

Calculus

>

Integrals

Medium

Applied

The value of $I = \displaystyle\int (x^3-1)^{1/3}x^5\,dx$ is

$\dfrac{(x^3-1)^{4/3}}{4} + \dfrac{(x^3-1)^{1/3}}{3} + C$ , where $C$ is an arbitrary constant

$\dfrac{(x^3-1)^{7/3}}{7} + \dfrac{(x^3-1)^{4/3}}{4} + C$ , where $C$ is an arbitrary constant

$\dfrac{(x^3-1)^{3/7}}{7} + \dfrac{(x^3-1)^{3/4}}{4} + C$ , where $C$ is an arbitrary constant

$\dfrac{(x^3-1)^{3/7}}{4} + \dfrac{(x^3-1)^{3/4}}{7} + C$ , where $C$ is an arbitrary constant

Correct Answer

Option 2

Correct Answer

Explanation

Q9:

2026: 21 May Shift 1

Calculus

>

Integrals

Easy

Common

The value of $\int_{-2}^{3} |x-1|\, dx$ is

5/2

-5/2

13/2

13

Correct Answer

Option 3

Correct Answer

Explanation

Q10:

2026: 21 May Shift 1

Calculus

>

Integrals

Medium

Common

$\int e^x \left(\frac{x-1}{2x^2}\right) dx$ is equal to ( where $C$ is an arbitrary constant)

$\frac{e^x}{2x^2} + C$

$-\frac{e^x}{2x^2} + C$

$\frac{e^x}{2x} + C$

$-\frac{e^x}{2x} + C$

Correct Answer

Option 3

Correct Answer

Explanation

Q11:

2026: 21 May Shift 1

Calculus

>

Integrals

Easy

Core

$\int \frac{dx}{\sin^2 x \cos^2 x} =$

$\tan x - \cot x + C$ : Where $C$ is an arbitrary constant

$-x + C$ : Where $C$ is an arbitrary constant

$x + C$ : Where $C$ is an arbitrary constant

$\tan x + \cot x + C$ : Where $C$ is an arbitrary constant

Correct Answer

Option 1

Correct Answer

Explanation

Q12:

2026: 21 May Shift 1

Calculus

>

Integrals

Medium

Core

Value of $\int (e^{2\log x} + e^{x\log 2} + e^{2\log 2})dx$ is

$\frac{x^3}{3} + 2^x \log 2 + 4x + C$ : where $C$ is an arbitrary constant

$\frac{x^3}{3} + \frac{2^x}{\log 2} + 4x + C$ : where $C$ is an arbitrary constant

$\frac{x^3}{3} + 2^x \log 2 + C$ : where $C$ is an arbitrary constant

$\frac{x^3}{3} + \frac{2^x}{\log 2} + C$ : where $C$ is an arbitrary constant

Correct Answer

Option 2

Correct Answer

Explanation

Q13:

2026: 21 May Shift 1

Calculus

>

Integrals

Medium

Core

Value of $\int_{1}^{4} \frac{\log_e x}{x^2}\, dx$ is

$3 - \log_e 4$

$\frac{1}{4}(3 - \log_e 4)$

$\frac{1}{4}\log_e 4$

$\frac{1}{4}\log_e \frac{1}{4}$

Correct Answer

Option 2

Correct Answer

Explanation

Q14:

2026: 21 May Shift 1

Calculus

>

Integrals

Easy

Applied

The integral $I = \int \frac{(1+\log x)^3}{x}\, dx$ equals:

$\frac{1}{x^4} + C$ , Where $C$ is an arbitrary constant

$\frac{(1+\log x)^4}{4} + C$ , Where $C$ is an arbitrary constant

$\frac{\log x}{x^3} + C$ , Where $C$ is an arbitrary constant

$\frac{(1+\log x)^3}{3} + C$ , Where $C$ is an arbitrary constant

Correct Answer

Option 2

Correct Answer

Explanation

Q15:

2026: 19 May Shift 2

Calculus

>

Integrals

Easy

Common

$\int 5^x dx$ is

(where C is an arbitrary constant)

$5^x \log_e 5 + C$

$5^x \log_e x + C$

$\frac{5^x}{\log_e x} + C$

$\frac{5^x}{\log_e 5} + C$

Correct Answer

Option 4

Correct Answer

Explanation

Q16:

2026: 19 May Shift 2

Calculus

>

Integrals

Medium

Core

$\int_1^4 \frac{\{x\}}{[x]} dx$ is equal to

[Where $\{x\}$ = fractional part function and $[x]$ = greatest integer function]

$\frac{1}{12}$

$\frac{11}{12}$

$\frac{2}{3}$

$\frac{4}{9}$

Correct Answer

Option 2

Correct Answer

Explanation

Q17:

2026: 19 May Shift 2

Calculus

>

Integrals

Medium

Core

$\int \frac{\log x - 3}{(\log x)^4} dx$ is equal to

(where C is an arbitrary constant and $\log_e x = \log x$ )

$\frac{x}{\log x} + C$

$\frac{x}{(\log x)^3} + C$

$\frac{1}{(\log x)^2} + C$

$\frac{x}{(\log x)^2} + C$

Correct Answer

Option 2

Correct Answer

Explanation

Q18:

2026: 19 May Shift 2

Calculus

>

Integrals

Easy

Applied

The integral $I = \int \frac{(1+\log_e x)^3}{x} dx$ is equal to

$\frac{x^4}{4} + C$ , where C is an arbitrary constant

$\frac{(1+x)^3}{3} + C$ , where C is an arbitrary constant

$\frac{(1+\log_e x)^4}{4} + C$ , where C is an arbitrary constant

$\frac{\log_e x}{4} + C$ , where C is an arbitrary constant

Correct Answer

Option 3

Correct Answer

Explanation

Q19:

2026: 19 May Shift 1

Calculus

>

Integrals

Medium

Common

$\int \frac{dx}{(a^2-x^2)^{3/2}}$ is equal to: (where C is an arbitrary constant)

$\frac{1}{a^2}\sqrt{a^2-x^2} + C$

$\frac{x}{a^2\sqrt{a^2-x^2}} + C$

$\frac{x}{a^2\sqrt{a^2+x^2}} + C$

$\frac{2}{5a^2}(a^2-x^2)^{5/2} + C$

Correct Answer

Option 2

Correct Answer

Explanation

Q20:

2026: 19 May Shift 1

Calculus

>

Integrals

Medium

Common

The value of integral $I = \int_{3}^{5} \frac{x^2}{(x-1)(x-2)}\,dx$ is

$2 - \log 2 + 4\log 3$

$\log 2 + 4\log 3$

$3 - \log 2 + \log 3$

$2 - \log 2 + \log 3$

Correct Answer

Option 1

Correct Answer

Explanation

Q21:

2026: 19 May Shift 1

Calculus

>

Integrals

Easy

Core

The value of $\int_{0}^{1} \frac{e^x}{1+e^{2x}}\,dx$ is

$\tan e - \frac{\pi}{4}$

$\tan^{-1}e - \frac{\pi}{4}$

$\tan^{-1}e - \frac{\pi}{2}$

$\tan^{-1}e - \frac{\pi}{3}$

Correct Answer

Option 2

Correct Answer

Explanation

Q22:

2026: 19 May Shift 1

Calculus

>

Integrals

Medium

Core

Match List-I with List-II

[c is an arbitrary constant]

List-I	List-II
(A) $\int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x}\,dx$	(I) $\tan x - x + c$
(B) $\int \frac{\sec^2 x}{\text{cosec}^2 x}\,dx$	(II) $2\tan x - 3\sec x + c$
(C) $\int \sec x(\sec x + \tan x)\,dx$	(III) $\tan x + \cot x + c$
(D) $\int \frac{2-3\sin x}{\cos^2 x}\,dx$	(IV) $\tan x + \sec x + c$

Choose the correct answer from the options given below:

(A) - (I), (B) - (IV), (C) - (II), (D) - (III)

(A) - (III), (B) - (II), (C) - (I), (D) - (IV)

(A) - (III), (B) - (I), (C) - (IV), (D) - (II)

(A) - (IV), (B) - (III), (C) - (I), (D) - (II)

Correct Answer

Option 3

Correct Answer

Explanation

Q23:

2026: 19 May Shift 1

Calculus

>

Integrals

Hard

Core

$\int \frac{x^9}{(4x^2+1)^6}\,dx$ is equal to:

[where c is an arbitrary constant]

$\frac{1}{5x}\left(4+\frac{1}{x^2}\right)^{-5} + c$

$\frac{1}{5}\left(4+\frac{1}{x^2}\right)^{-5} + c$

$\frac{1}{10x}(1+4)^{-5} + c$

$\frac{1}{10}\left(\frac{1}{x^2}+4\right)^{-5} + c$

Correct Answer

Option 4

Correct Answer

Explanation

Q24:

2026: 19 May Shift 1

Calculus

>

Integrals

Easy

Core

The value of the integral $\int_{-6}^{0} |x+3|\,dx$ is

9/4

1/2

9

9/2

Correct Answer

Option 3

Correct Answer

Explanation

Q25:

2026: 19 May Shift 1

Calculus

>

Integrals

Easy

Applied

Match List-I with List-II

List-I (Definite Integral)	List-II (Value)
(A) $\int_{1}^{2} e^x\,dx$	(I) $\frac{e^2(e^2-1)}{2}$
(B) $\int_{0}^{1} e^x\,dx$	(II) $\frac{(e^2-1)}{2}$
(C) $\int_{1}^{2} e^{2x}\,dx$	(III) $e-1$
(D) $\int_{0}^{1} e^{2x}\,dx$	(IV) $e(e-1)$

Choose the correct answer from the options given below:

(A) - (IV), (B) - (III), (C) - (I), (D) - (II)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (I), (B) - (II), (C) - (IV), (D) - (III)

Correct Answer

Option 1

Correct Answer

Explanation

Q26:

2026: 19 May Shift 1

Calculus

>

Integrals

Medium

Applied

The value of $\int_{0}^{3} \frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}}\,dx$ is

$\frac{1}{2}$

1

$\frac{3}{2}$

2

Correct Answer

Option 3

Correct Answer

Explanation

Q27:

2026: 18 May Shift 2

Calculus

>

Integrals

Easy

Common

$\int 3x^2 e^{x^3}\,dx$ is equal to : (where $C$ is an arbitrary constant)

$3x^3+C$

$\dfrac{1}{3}e^{x^3}+C$

$e^{x^3}+C$

$\dfrac{1}{2}e^{x^2}+C$

Correct Answer

Option 3

Correct Answer

Explanation

Q28:

2026: 18 May Shift 2

Calculus

>

Integrals

Medium

Common

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(x)+f(y)=f\left(\dfrac{x+y}{1-xy}\right)$ , Where $\mathbb{R}$ is set of real numbers then $\alpha+\int_{-\alpha}^{\alpha} f(x)\,dx$ is equal to:

$\alpha$

$2\alpha$

0

$\alpha - 1$

Correct Answer

Option 1

Correct Answer

Explanation

Q29:

2026: 18 May Shift 2

Calculus

>

Integrals

Medium

Core

The value of $\int_{-1}^{1} \dfrac{x^6}{6^x+1}\,dx$ is

3

6

$\dfrac{2}{7}$

$\dfrac{1}{7}$

Correct Answer

Option 4

Correct Answer

Explanation

Q30:

2026: 18 May Shift 2

Calculus

>

Integrals

Medium

Core

If $f(x)=\int e^x(\tan x+\tan^2 x)\,dx$ and $f(0)=0$ , then $f\left(\dfrac{\pi}{4}\right)$ equals:

0

-1

1

$e^{\pi/4}$

Correct Answer

Option 3

Correct Answer

Explanation

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